© Revista de Matemática: Teoría y Aplicaciones 2022 29(1) : 1–17 cimpa – ucr issn: 1409-2433 (Print), 2215-3373 (Online)

doi: https:// doi. org/ 10.15517/ rmta. v29i1. 43782

lie group symmetries’ complete classification for a generalized chazy

equation and its equivalence group

clasificación completa del grupo de simetrías de lie para una ecuación de chazy generalizada y su grupo de equivalencia

Óscar M. Londoño Duque^{∗}  Yeison A. Acevedo^{†}

Gabriel I. Loaiza^{‡}

Received: 21/Oct/2020; Revised: 3/Sep/2021; Accepted: 22/Nov/2021

∗IMECC-UNICAMP, Instituto de Matemáticas, Campinas, Brasil. E-Mail: o154278@dac.unicamp.br

†EAFIT, Departamento de Ciencias Matemáticas, Medellín, Colombia. E-Mail: yaceved2@eafit.edu.co

‡EAFIT, Departamento de Ciencias Matemáticas, Medellín, Colombia. E-Mail: gloaiza@eafit.edu.co

Abstract

In this work, a complete classification of the Lie group symmetries for a generalization of Chazy equation was carried out and the equivalence group for the generalized Chazy equation is calculated and used to present the principal algebra of the equation.

Keywords: Lie symmetries; equivalence group; Lie symmetries classification; Chazy generalized equation .

Resumen

En este trabajo se obtiene una clasificación completa del grupo de simetrías de Lie para una generalización de la ecuación de Chazy, se calcula el grupo de equivalencia y se utiliza éste para presentar el álgebra principal de la ecuación.

Palabras clave: simetrías de Lie; grupo de equivalencia; clasificación de simetrías de Lie; ecuación generalizada de Chazy.

Mathematics Subject Classification: 76M60, 70G65, 34C14

1 Introduction

In [3], [8], [12], [9] it is introduced Prandtl’s boundary layer equation for the stream function for an incompressible, steady two-dimensional flow with uniform or vanishing mainstream velocity as

νu_{yyy} = u_{y}u_{xy} − u_{x}u_{yy},  (1)

where ν is a real number and represent the kinematic viscosity, the mentioned authors also introduce the following similarity transformation

u(x, y) = x^{1−α}g(ω), ω = \frac{y}{x^{α}} ,  (2)

where α is a real number. These authors affirm that using (2) in (1) the following non-linear third-order differential equation is obtained

νg_{ωωω} + Dgg_{ω} + A(g_{ω})^{2} = 0,  (3)

where g(ω) is a real function and at least C^{(3)} and D = 1 − α (2-dimensional form) or D = 2 − α (radial form), with A = 2 − α in both cases. In [9], the similarity transformation was made explicitly, that is, the author details all the calculations.

In [3], [12], a rough historical approach in different areas of mathematics and physics is presented for the equation (3).

In [27], Schlichting proposes a numerical solution for a free 2-dimensional jet in which α = 2/3 and later an analytic solution was derived by Bickley [4]. In [28], Squire obtained the solution for the free radial jet for which α = 1. In [13], Glauert presents solutions in parametric form when α = \frac{5}{4} (2-dimensional) and α = \frac{3}{4} (radial), and in [26], Riley presents a solution with α = 2 (radial).

The case with α = −1 (2-dimensional), α = −4 (radial) and ν = 1 has been the most studied equation using the theory of symmetries. In this case (3) becomes

y_{xxx} − 2yy_{xx} + 3y_{x}^{2} = 0 .  (4)

This equation, known as Chazy equation, was introduced by the same author in [10]. On the other hand, continuing with the use of Lie symmetries for the equation (4), in [11] Chazy established that the group of symmetries of (4) is a nonsolvable group, and determined a reduction of this equation. In [12] Clarkson and Olver also established that the group of symmetries of (4) is a non-solvable algebra, and expressed the general solution of (4) as the ratio of two solutions of a hypergeometric equation. In [17], Ibragimov and Nucci, using the theory of Lie symmetries, were able to reduce (4) by the method of semi-canonical variables. In [1] the theory of non-local Lie symmetries was used to reduce (4), and later in [23] an improvement was presented. In [2], Arrigo calculated the group of symmetries and the invariant transformation of this group using canonical variables. Recently, in [20], [22], the authors characterized invariant solutions for the equation (4) and Kummer-Schwarz equation, from the generator operators of the optimal system, and using it, they presented new solutions.

In this line, given a differential equation, we can be considered by giving a generalization of it, defining coefficients from a differential equation in terms of parameters, obtained a family of EDS. Within this context, a problem of interest is to classify said family, into equivalence classes, according to the equivalence relation that defines each pair of differential equations of the family as equivalent if and only if, both differential equations have the same group of Lie symmetries. In this way, the family is divided into equivalence classes and is called a complete classification of Lie symmetries group (either the description of the classes or the description of the respective groups of symmetries), this allows knowing all the differential equations of the family that have the same group of Lie symmetries for the initial equation.

Besides, an equivalence transformation for a family the differential equations is a transformation of change of variables, which makes each member of the family correspond to a differential equation equivalent. The set of equivalence’s transformations, endowed with the binary operation defined by the Lie bracket, is called equivalence group of the differential equations family. This type of study has been carried out in [23] for the Chazy equation, considering the coefficients as constant functions.

In this work, we extend the discussion started in [23], considering the following generalization of (4):

y_{xxx} − f(y)y_{xx} + 3y_{x}^{2} = 0,  (5)

where f is an arbitrary function. Then, our objectives are: i) to provide a complete classification of Lie symmetries group for (5), ii) to determine the equivalence group of (5), and finally, iii) to illustrate the application of the equivalence group to obtain principal algebra.

2 Lie symmetries

In this section, we study the Lie point symmetries of (5). Following the classical Lie technique for calculating the symmetries of differential equations [5, 7, 18, 25], we carried out the complete group classification for (5). The corresponding result can be stated as follows:

Proposition 1 The Lie point symmetry group of the generalized Chazy equation (5) with arbitrary f, is generated by

X_{1} = ∂_{x}. Principal algebra,  (6)

other complementary cases are presented in the Table 1.

Proof. A general form of the one-parameter Lie group admitted by (5) is given by

x → x + ϵξ(x, y) + ⋯ and y → y + ϵη(x, y) + ⋯,

where ϵ is the group parameter. The vector field associated with the group of transformations shown above can be written as Γ = ξ(x, y)\frac{∂}{∂x} + η(x, y)\frac{∂}{∂y} . Applying its third prolongation

Γ^{(3)} = Γ + η_{[x]} \frac{∂}{∂y_{x}} + η_{[xx]} \frac{∂}{∂y_{xx}} + η_{[xxx]} \frac{∂}{∂y_{xxx}} ,

Table 1: Infinitesimal generators of (5).

Case  f(y)  Condition Infinitesimal generators of the group
o)  f(y)  f_{yy}(y) ≠ 0  X_{1} = ∂_{x} (Principal algebra ).
i) f(y) = yk_{1} + k_{2} k_{1}, k_{2} ≠ 0  X_{1}, X_{2} = −x∂_{x} + (y + \frac{k_{2}}{k_{1}} ) ∂_{y}.
ii) f(y) = 2y + k_{2}  k_{2} ≠ 0  \begin{matrix}X_{1}, X_{2} = −x∂_{x} + (y + \frac{k_{2}}{2} ) ∂_{y}, \\ X_{3} = −\frac{x^{2}}{2} ∂_{x} + (yx + \frac{k_{2}}{2} + 3) ∂_{y}. \end{matrix}

iii)  f(y) = 2y  - - -  \begin{matrix}X_{1}, X_{2} = −x∂_{x} + y∂_{y}, \\ X_{3} = −\frac{x^{2}}{2} ∂_{x} + (yx + 3)∂_{y}. \end{matrix}
iv)  f(y) = k_{2}  k_{2} ≠ 0  X_{1}, X_{2} = ∂y.
v)  f(y) = yk_{1}  k_{1} ≠ 0, 2  X_{1}, X_{2} = x∂_{x} − y∂_{y}.

to (5), we must find the infinitesimals ξ(x, y) , η(x, y) satisfying the symmetry condition

η(−f_{y}(y)y_{xx}) + η_{[x]}(6y_{x}) + η_{[xx]}(−f(y)) + η_{[xxx]} = 0,  (7)

associated with (5). Here, η_{[x]}, η_{[xx]} and η_{[xxx]} are the coefficients in Γ^{(3)} given by

η_{[x]} = D_{x}[η] − (D_{x}[ξ])y_{x} = η_{x} + (η_{y} − ξ_{x})y_{x} − ξ_{y}y_{x}^{2},
η_{[xx]} = D_{x}[η_{[x]}] − (D_{x}[ξ])y_{xx}
= η_{xx} + (2η_{xy} − ξ_{xx})y_{x} + (η_{yy} − 2ξ_{xy})y_{x}^{2} − ξ_{yy}y_{x}^{3}
+(η_{y} − 2ξ_{x})y_{xx} − 3ξ_{y}y_{x}y_{xx}
and η_{[xxx]} = D_{x}[η_{[xx]}] − (D_{x}[ξ])y_{xxx}  (8)
= η_{xxx} + (3η_{xxy} − ξ_{xxx})y_{x} + 3(η_{xyy} − ξ_{xxy})y_{x}^{2}
+ (η_{yyy} − 3ξ_{xyy}) y_{x}^{3} − ξ_{yyy}y_{x}^{4} + 3 (η_{xy} − ξ_{xx}) y_{xx}
+3 (η_{yy} − 3ξ_{xy}) y_{x}y_{xx} − 6ξ_{yy}y_{x}^{2}y_{xx} − 3ξ_{y}y_{xx}^{2}
+ (η_{y} − 3ξ_{x}) y_{xxx} − 4ξ_{y}y_{x}y_{xxx};

where D_{x} is the total derivative operator: D_{x} = ∂_{x}+y_{x}∂_{y}+y_{xx}∂_{y_{x}}+y_{xxx}∂_{y_{xx}} · · · . Replacing (8) into (7) we obtain:

η(−f_{y}(y)y_{xx}) + (6y_{x})(η_{x} + (η_{y} − ξ_{x})y_{x} − ξ_{y}y_{x}^{2})
−(f(y))(η_{xx} + (2η_{xy} − ξ_{xx})y_{x} + (η_{yy} − 2ξ_{xy})y_{x}^{2} − ξ_{yy}y_{x}^{3})
−(f(y))((η_{y} − 2ξ_{x})y_{xx} − 3ξ_{y}y_{x}y_{xx})
+η_{xxx} + (3η_{xxy} − ξ_{xxx})y_{x} + 3(η_{xyy} − ξ_{xxy})y_{x}^{2}
+ (η_{yyy} − 3ξ_{xyy}) y_{x}^{3} − ξ_{yyy}y_{x}^{4} + 3 (η_{xy} − ξ_{xx}) y_{xx}
+3 (η_{yy} − 3ξ_{xy}) y_{x}y_{xx} − 6ξ_{yy}y_{x}^{2}y_{xx} − 3ξ_{y}y_{xx}^{2}
+ (η_{y} − 3ξ_{x}) y_{xxx} − 4ξ_{y}y_{x}y_{xxx} = 0.

If we denote f ≅ f(y) and substitute y_{xxx} = fy_{xx} − 3y_{x}^{2} in the last expression, then we can rearrange the expression with respect to 1, y_{x}, y_{x}^{2}, y_{x}^{3}, y_{x}^{4}, y_{x}y_{xx}, y_{x}^{2}y_{xx}, y_{xx} and y_{xx}^{2} for obtain

η_{xxx} − fη_{x} + (6η_{x} − f(2η_{xy} − ξ_{xx}) + 3η_{xxy} − ξ_{xxx})y_{x}
(6η_{y} − 6ξ_{x} − f(η_{yy} − 2ξ_{xy}) + 3η_{xyy} − 3ξ_{xxy} − 3(η_{y} − 3ξ_{x}))y_{x}^{2}
+(−6ξ_{y} + fξ_{yy} + η_{yyy} − 3ξ_{xyy} + 12ξ_{y})y_{x}^{3} − ξ_{yyy}y_{x}^{4}
+(3fξ_{y} + 3η_{yy} − 9ξ_{xy} − 4fξ_{y})y_{x}y_{xx} − 6ξ_{yy}y_{x}^{2}y_{xx}
+(−f_{y}η − f(η_{y} − 2ξ_{x}) + 3η_{xy} − 3ξ_{xx} + f(η_{y} − 3ξ_{x}))y_{xx}
−3ξ_{y}y_{xx}^{2} = 0.

As we know, the variables 1, y_{x}, y_{x}^{2}, y_{x}^{3}, y_{x}^{4}, y_{x}y_{xx}, y_{x}^{2}y_{xx}, y_{xx} and y_{xx}^{2}, are linearly independent, thus the previous expression we obtain the determining equations

ξ_{y} = η_{yy} =0,  (9a)
η_{y} + ξ_{x} =0,  (9b)
−fη_{xx} + η_{xxx} =0,  (9c)
f(−2η_{xy} + ξ_{xx}) + 6η_{x} + 3η_{xxy} − ξ_{xxx} =0,  (9d)
−f_{y}η − fξ_{x} + 3η_{xy} − 3ξ_{xx} =0.  (9e)

Solving in (9a), we have ξ = c_{1}(x) and η = yc_{2}(x) + c_{3}(x), with c_{1}, c_{2}, c_{3} as arbitrary functions. From the expressions for ξ and η into (9b), (9c), (9d) and (9e), we obtain respectively:

c_{2}(x) + c′_{1}(x) =0, (10a)
−(yc′′_{2}(x) + c′′_{3}(x))f + yc′′′_{2}(x) + c′′′_{3}(x) =0, (10b)
[−2c′_{2}(x) + c′′_{1}(x)]f + 6[yc′_{2}(x) + c′_{3}(x)] + 3c′′_{2}(x) − c′′′_{1}(x) =0, (10c)
−[yc_{2}(x) + c_{3}(x)]f_{y} − c′_{1}(x)f + 3c′_{2}(x) − 3c′′_{1}(x) =0. (10d)

Using (10a) and differentiating (10c) with respect to y we obtain −3c′_{2}(x)f_{y} + 6c′_{2}(x) = 0, whose derivative with respect to y yields

− 3c′_{2}(x)f_{yy} = 0.  (11)

From (11), we consider two cases: f_{yy} ≠ 0 or f_{yy} = 0.

First, we consider the case f_{yy} ≠ 0. According to (11), we can assert that −3c′_{2}(x) = 0, hence c_{2}(x) = k_{1}, with k_{1} an arbitrary constant. Then from (10a), we obtain:

c_{2}(x) = k_{1},  c_{1}(x) = −xk_{1} + k_{2},  (12)

where k_{2} is an arbitrary constant. By replacing (12) into (10c), we get c′_{3}(x) = 0, so that c_{3}(x) = k_{3}. From the above and (12) into (10d), it follows that −(yk_{1} + k_{3})f_{y} + f(k_{1}) = 0. By differentiating in the previous expression with respect to y, we have (yk_{1} + k_{3})f_{yy} = 0. But we know that f_{yy} ≠ 0, therefore k_{1} = 0 and k_{3} = 0. Consecuently, from c_{3}(x) = k_{3} and (12), we obtain the infinitesimal generators of the group, ξ = c_{1}(x) = k_{2} and η = k_{1}y + k_{3} = 0, which is equivalent to X_{1} = ∂_{x} with f_{yy} ≠ 0.

We consider now the case f_{yy} = 0, and proceed to use (11) to establish (6). First, from f_{yy} = 0 we have

f(y) = yk_{1} + k_{2},  (13)

with k_{1}, k_{2} as arbitrary constants. From (13), we see that there are two cases to be considered regarding k_{1} ≠ 0 or k_{1} = 0.

1. Case k_{1} ≠ 0. For (13) consider two cases: k_{2} ≠ 0 or k_{2} = 0.

(a) Case k_{1} ≠ 0 and k_{2} ≠ 0. From (13) and (10b) we get −(yk_{1} + k_{2})(yc′′_{2}(x) + c′′_{3}(x)) + yc′′′_{2}(x) + c′′′_{3}(x) = 0. By differentiating the last expression with respect to y, we obtain

− k_{2}c′′_{2}(x) − 2k_{1}yc′′_{2}(x) − k_{1}c′′_{3}(x) + c′′′_{2}(x) = 0. (14)

Now, by differentiating (14) with respect to y, we find −2k_{1}c′′_{2}(x) = 0, that implies c′_{2}(x) = k_{3}, and thus, c_{2}(x) = xk_{3} + k_{4}, where k_{3}, k_{4} are arbitrary constants. From (10a) we have

c_{1}(x) = −\frac{x^{2}}{2} k_{3} − xk_{4} + k_{5}.  (15)

By substitution of c_{2}(x) = xk_{3} + k_{4} into (14), we get c′′_{3}(x) = 0, so

c_{3}(x) = xk_{6} + k_{7}.  (16)

From (13), c_{2}(x) = xk_{3} + k_{4}, (15) and (16) into (10c), we obtain

(yk_{1} + k_{2})(−3k_{3}) + 6(yk_{3} + k_{6}) = 0,  (17)

and by differentiating this expression with respect to y, we find

k_{3}(−k_{1} + 2) = 0.  (18)

In last equation we see that there are two cases to be considered: k_{3} = 0 or k_{3} ≠ 0.

i. Case k_{1} ≠ 0, k_{2} ≠ 0, and k_{3} = 0. From (17) we have k_{6} = 0. From c_{2}(x) = k_{4}, (15) and (16), we have:

c_{1}(x) = −xk_{4} + k_{5}, c_{2}(x) = k_{4}, c_{3}(x) = k_{7}.  (19)

From (19) and (13) into (10d), we find −k_{1}k_{7} + k_{2}k_{4} = 0, implies k_{7} = \frac{k_{2}k_{4}}{k_{1}} . Then, we can rewrite (19) as :

c_{1}(x) = −xk_{4} + k_{5}, c_{2}(x) = k_{4}, c_{3}(x) = \frac{k_{2}k_{4}}{k_{1}} .

Summarizing, the infinitesimal generators of the group are: η = yk_{4} + \frac{k_{2}k_{4}}{k_{1}} , ξ = −xk_{4} + k_{5}, which is equivalent to:

X_{1} = ∂_{x} and X_{2} = −x∂_{x} + (y + \frac{k_{2}}{k_{1}} ) ∂_{y}.

ii. Case k_{1} ≠ 0, k_{2} ≠ 0 and k_{3} ≠ 0. From (18), we have k_{1} = 2. From (17), (2y + k_{2})(−3k_{3}) + 6(yk_{3} + k_{6}) = 0, which implies k_{6} = \frac{k_{3}k_{2}}{2} . From the above, c_{2}(x) = xk_{3} + k_{4}, (15) and (16), we obtain

c_{1}(x) = −\frac{x^{2}}{2} k_{3} − xk_{4} + k_{5}, c_{2}(x) = xk_{3} + k_{4},
c_{3}(x) = x\frac{k_{3}k_{2}}{2} + k_{7}.  (20)

Inserting (13) and (20) into (10d), we find −2[y(xk_{3} + k_{4}) + (x\frac{k_{3}k_{2}}{2} + k_{7})] − (2y + k_{2})[−(xk_{3} + k_{4})] + 6k_{3} = 0, hence k_{7} = \frac{k_{2}k_{4}}{2} + 3k_{3}. From this into (20) it follows that c_{1}(x) = −\frac{x^{2}}{2} k_{3} − xk_{4} + k_{5}, c_{2}(x) = xk_{3} + k_{4}, c_{3}(x) = x\frac{k_{3}k_{2}}{2} + \frac{k_{2}k_{4}}{2} + 3k_{3}.

Summarizing, the infinitesimal generators of the group are:

η = yxk_{3} + yk_{4} + x\frac{k_{3}k_{2}}{2} + \frac{k_{2}k_{4}}{2} + 3k_{3},

ξ = −\frac{x^{2}}{2} k_{3} − xk_{4} + k_{5},

which is equivalent to X_{1} = ∂_{x}, X_{2} = −x∂_{x} + (y + \frac{k_{2}}{2} )∂_{y}, and

X_{3} = −\frac{x^{2}}{2} ∂_{x} + (yx + \frac{k_{2}}{2} + 3)∂_{y}.

(b) Case k_{1} ≠ 0 and k_{2} = 0. Since (13), we have f(y) = yk_{1}, then using (10b), we have −y^{2}(k_{1}c′′_{2}(x)) − y(k_{1}c′′_{3}(x)) − c′′′_{2}(x)) + c′′′_{3}(x) = 0. By differentiating this with respect to y, it follows

− 2yk_{1}(c′′_{2}(x)) − k_{1}c′′_{3}(x) + c′′′_{2}(x) = 0.  (21)

From this, we conclude that c′′_{2}(x) = 0, thus c_{2}(x) = xk_{3} + k_{4}, with k_{3}, k_{4} as arbitrary constants. When replacing c_{2}(x) = xk_{3} + k_{4} into (21), we obtain c′′_{3}(x) = 0, hence c_{3}(x) = xk_{6} + k_{7}, with k_{6}, k_{7} as arbitrary constants. From c_{2}(x) = xk_{3} + k_{4} into (10a), we find c′_{1}(x) = −xk_{3} − k_{4}, hence c_{1}(x) = −\frac{k_{3}}{2} x^{2} − xk_{4} + k_{5}. Now, from f(y) = yk_{1}, c_{2}(x) = xk_{3} + k_{4}, c_{3}(x) = xk_{6} + k_{7} and c_{1}(x) = −\frac{k_{3}}{2} x^{2} − xk_{4} + k_{5} into (10c), we obtain yk_{3}(−k_{1} + 2) + 2k_{6} = 0, which implies that

k_{6} = 0  and  k_{3}(−k_{1} + 2) = 0.  (22)

Then, from (22) consider two cases: k_{3} = 0 or k_{3} ≠ 0.

i. Case k_{1} ≠ 0, k_{2} = 0 and k_{3} = 0. From c_{2}(x) = xk_{3} + k_{4}, c_{3}(x) = xk_{6} + k_{7} and c_{1}(x) = −\frac{k_{3}}{2} x^{2} − xk_{4} + k_{5}, we have

c_{1}(x) = −xk_{4} + k_{5}, c_{2}(x) = k_{4} and c_{3}(x) = k_{7}.

From the last expressions into (10d), we obtain k_{7} = 0.

Then, we get: c_{1}(x) = −xk_{4} + k_{5}, c_{2}(x) = k_{4} and c_{3}(x) = 0. Summarizing, the infinitesimal generators of the group are: η = yk_{4} and ξ = −xk_{4} + k_{5}, which is equivalent to:

X_{1} = ∂_{x} and X_{2} = x∂_{x} − y∂_{y}.

ii. Case k_{1} ≠ 0, k_{2} = 0 and k_{3} ≠ 0. From (22), we have k_{1} = 2. Then, from c_{2}(x) = xk_{3} + k_{4}, c_{3}(x) = xk_{6} + k_{7}, and c_{1}(x) = −\frac{k_{3}}{2} x^{2}−xk_{4}+k_{5}, we find: c_{1}(x) = −\frac{k_{3}}{2} x^{2}−xk_{4}+k_{5}, c_{2}(x) = xk_{3} + k_{4} and c_{3}(x) = k_{7}. Summarizing, the infinitesimal generators of the group are: η = yxk_{3} + yk_{4} + k_{7}, ξ = −\frac{k_{3}}{2} x^{2} − xk_{4} + k_{5}, with k_{7} = 3k_{3}, which is equivalent to:

X_{1} = ∂_{x}, X_{2} = −x∂_{x} + y∂_{y}, X_{3} = −\frac{x^{2}}{2} ∂_{x} + (yx + 3)∂_{y} .

2. From (13), suppose that k_{1} = 0. Then, we have f(y) = k_{2} and consequently, from (10b) we obtain:

y[−k_{2}c′′_{2}(x) + c′′′_{2}(x)] − k_{2}c′′_{3}(x) + c′′′_{3}(x) = 0.

Using (10a) in the expression above we can conclude:

c_{1}(x) = \frac{−a_{1}}{k_{2}^{3}} e^{xk_{2}} − \frac{a_{3}}{2} x^{2} − xa_{2} + a_{4}, c_{2}(x) = \frac{a_{1}}{k_{2}^{2}} e^{xk_{2}} + xa_{3} + a_{2},

c_{3}(x) = \frac{a_{5}}{k_{2}^{2}} e^{xk_{2}} + xa_{6} + a_{7},  (23)

with a_{1}, ⋯, a_{7} arbitrary constants. Using (23) into (10c) we get:

e^{xk_{2}} [y (\frac{6a_{1}}{k_{2}} ) + a_{1} + \frac{6a_{5}}{k_{2}} ] + 6a_{6} − 3a_{3}k_{2} + y(6a_{3}) = 0,

which implies that a_{1} = a_{3} = a_{5} = a_{6} = 0. From (23), c_{1}(x) = −xa_{2} + a_{4}, c_{2}(x) = a_{2} and c_{3}(x) = a_{7}. From the last expressions into (10d), we have k_{2}a_{2} = 0, hence a_{2} = 0. Then, c_{1}(x) = a_{4}, c_{2}(x) = 0, and c_{3}(x) = a_{7}. Thus, the infinitesimal generators of the group are η = a_{7} and ξ = a_{4}, which is equivalent to

X_{1} = ∂_{x} and X_{2} = ∂_{y}.

Remark 1 In [10], the generators of the symmetry group of Equation (4) are presented without showing details for the calculations, these details are presented in [20], where the optimal algebra and new invariant solutions are presented too. Now, the previous results for the generators of the Lie symmetries group coincide with the part (iii) of Proposition 1, after manipulating some constants, which are: X_{1} = ∂_{x}, X_{2} = x∂_{x} − y∂_{y} and X_{3} = x^{2}∂_{x} − (2yx + 6)∂_{y}.

3 The equivalence group

An equivalence transformation for (5) is a change of variables (x, y) → (x̃, ỹ), where the structure of (5) is conserved. The infinitesimal generators of the continuous group of equivalence transformations have the form

Γ = ξ(x, y)∂_{x} + η(x, y)∂_{y} + μ(x, y, f)∂_{f}.  (24)

According to [25, 15], the operator (24) generates the continuous equivalence group if it is admitted by the extended system:

\begin{cases}y_{xxx} − fy_{xx} + 3y_{x}^{2} = 0, \\ f_{x} = 0. \end{cases}  (25)

The infinitesimal invariance test for the extended system (25) requires the following prolongation of the operator (24),

Γ^{(3)} = ξ(x, y)∂_{x} + η(x, y)∂_{y} + μ(x, y, f)∂_{f}
+η_{[x]}∂_{y_{x}} + η_{[xx]}∂_{y_{xx}} + η_{[xxx]}∂_{y_{xxx}} + ω∂_{f_{x}},

where η_{[x]}, η_{[xx]} and η_{[xxx]} are given in (8), ω = D̃_{x}[μ] − f_{x}D̃_{x}[ξ] − f_{y}D̃_{x}[η] and D̃_{x} = \frac{∂}{∂x} + f_{x} \frac{∂}{∂f} + f_{xx} \frac{∂}{∂f_{x}}. Applying Γ^{(3)} to the extended system (25), we obtain that the invariance condition:

\begin{cases}μ(−y_{xx}) + η_{[x]}(6y_{x}) + η_{[xx]}(−f) + η_{[xxx]} = 0, \\ ω = 0. \end{cases}  (26)

Substituting (25) in (26), we have ω = μ_{x} − f_{y}(η_{x}) = 0. Now, as the constant 1 and f_{y} are algebraically independent, then first equation above implies μ_{x} = 0 and η_{x} = 0. Thus μ = μ(y, f), η = η(y), and ξ = ξ(x, y). With these in hand, we substitute (8) into (26) and get

μ(−y_{xx}) + (6y_{x})((η_{y} − ξ_{x})y_{x} − ξ_{y}y_{x}^{2})
− (f)((−ξ_{xx})y_{x} + (η_{yy} − 2ξ_{xy})y_{x}^{2} − ξ_{yy}y_{x}^{3} + (η_{y} − 2ξ_{x})y_{xx} − 3ξ_{y}y_{x}y_{xx})
− (ξ_{xxx})y_{x} + 3(−ξ_{xxy})y_{x}^{2} + (η_{yyy} − 3ξ_{xyy}) y_{x}^{3} − ξ_{yyy}y_{x}^{4} + 3 (−ξ_{xx}) y_{xx}
+3 (η_{yy} − 3ξ_{xy}) y_{x}y_{xx} − 6ξ_{yy}y_{x}^{2}y_{xx} − 3ξ_{y}y_{xx}^{2}
+ (η_{y} − 3ξ_{x}) y_{xxx} − 4ξ_{y}y_{x}y_{xxx} = 0.

now, we replace y_{xxx} = y_{xx}f − 3y_{x}^{2} in the last expression.

As know, the variables y_{x}, y_{x}^{2}, y_{x}^{3}, y_{x}^{4}, y_{x}y_{xx}, y_{x}^{2}y_{xx}, y_{xx}, y_{xx}^{2} are linearly independent; hence we obtain the following system of determining equations

ξ_{y} = η_{yy} = μ_{x} = η_{x} =0,  (27a)
η_{y} + ξ_{x} =0,  (27b)
fξ_{xx} − ξ_{xxx} =0,  (27c)
μ + fξ_{x} + 3ξ_{xx} =0.  (27d)

From (27a), we get

ξ = c_{1}(x),  η = yc_{2} + c_{3}  and μ = μ(y, f),  (28)

where c_{1} is arbitrary function and c_{2}, c_{3} are arbitrary constants. From (28) into (27b) it follows c_{2} + c′_{1}(x) = 0, which implies c_{1}(x) = −c_{2}x + c_{4}. From the last expresion in (27d) we get μ = fc_{2}, then the infinitesimal generators of the equivalence group of the (5) are η = yc_{2} + c_{3}, ξ = −c_{2}x + c_{4}, and μ = fc_{2} which is equivalent to X_{1} = −x∂_{x} + y∂_{y} + f∂_{f}, X_{2} = ∂_{x} and X_{3} = ∂_{y}. We summarize the above considerations as follows:

Proposition 2 The Lie algebra for the continuous equivalence group of Equation (5), is generated by the following vector fields:

X_{1} = −x∂_{x} + y∂_{y} + f∂_{f},  X_{2} = ∂_{x},  X_{3} = ∂_{y}.  (29)

4 Principal algebra using the equivalence group

In this section, we show the consistency of the previous results by using the equivalence group to calculate the principal algebra, which coincides with that obtained in the Proposition 1.

The most general symmetries algebra of Equation (5), for an arbitrary function f is called the Principal Lie Algebra L_{𝔭} [15, 19]. This algebra can be obtained from equivalence algebra L_{𝔢} (29). Let us denote by x = (x) and y = (y) the independent and dependent variables, respectively, and by f ≡ {f} the arbitrary elements in the system (5). Let the projection X = pr_{(x,y)}(Y) of the (29) to the space (x, y) of the independent and dependent variables

X = pr_{(x,y)}(Y) = ξ∂x + η∂y.  (30)

Also, consider the projection Z = pr_{(y,f)}(Y) to the space (y, f) involved in the arbitrary elements

Z = pr_{(y,f)}(Y) = η∂y + μ∂f.  (31)

From the above, it is clear that operators X and Z are well defined, i.e. their coordinates involve only the respective variables (x, y) and (y, f). For our specific case, the Proposition on projections was formulated in [16, 19] as follows:

Proposition 3 An operator X belongs to the principal Lie algebra L_{𝔭} for (5) if and only if X = pr_{(x,y)}(Y), where Y is an equivalence generator such that Z = pr_{(y,f)}(Y) ≡ 0.

Now, the idea is to apply Proposition 3 to find the principal Lie algebra L_{𝔭} of (5). Let‘s consider the equivalence generator Y = k_{1}X_{1} + k_{2}X_{2} + k_{3}X_{3}, which is a linear combination of the operators in (29). The above is equivalent to

Y = k_{1}[−x∂x + y∂y + f∂f] + k_{2}[∂_{x}] + k_{3}[∂_{y}],  (32)

with projections (30) and (31) as

X = [−xc_{2} + c_{4}]∂x + [yc_{2} + c_{3}]∂y,
Z = [yc_{2} + c_{3}]∂y + [fc_{2}]∂f.

Now, we know that Z ≡ 0 if and only if c_{2} = c_{3} = 0, thus applying Proposition 3, we can conclude that Y = k_{2}X_{2}. So, the principal Lie algebra L_{𝔭} of (5) is spanned by {X_{2}}, where X_{2} is the presented in the Proposition 2 which coincides with (6), from Proposition 1.

Remark 2 Note that the Proposition 3 can be proved with the following considerations: We recall that the principal Lie algebra consists of all the operators Γ = ξ(x, y)\frac{∂}{∂x} + η(x, y)\frac{∂}{∂y} admitted by equation (5) for any f(y). Therefore the principal Lie algebra is the subalgebra of the equivalence algebra, such that any operator Y (in our case these operator are X_{1}, X_{2}, X_{3}) of this subalgebra leaves invariant equations f = f(y). It follows that f and y are invariant with respect to Y. It means that η = 0 and μ = 0, or P r(y, f)(Y) = 0.

5 Conclusions

In this work, the complete classification of the symmetry group of the generalized Chazy equation (5) was carried out (Proposition 1). According to Table 1, the family of differential equations defined by (5), where f(y) is a parameter, the only equation that has the same group of symmetries as the Chazy equation (4), is itself.

Furthermore, in the item ii) of the table 1 it can be noted that there is a more general Chazy equation that has the same group symmetries as the equation (4). In Proposition 1, the principal algebra is calculated and it coincides with that obtained by applying Proposition 3 to the equivalence group.

Ideas to develop in future works could be calculating the conservation laws for (4), using the symmetries presented in Proposition 1, also it is possible to investigate the contact symmetries as was presented in [24, 14, 6] and try to obtain for the symmetries associated with the equation (4), the classification of its Lie algebra as was presented in [21].

Acknowledgments

The author Y. Acevedo is grateful for the financial support of the EAFIT University for the Master’s Scholarship. G. Loaiza and Y. Acevedo also thank the Project of MinScience “Sobre procesos de difusión y simplificación de información” (code 121671250122). The anonymous referees, whose comments and suggestions helped to improve the paper, are gratefully acknowledged.

Financial support

EAFIT University, Science Math. Department, Colombia.

Data availability

The data used to support the findings of this study are included within the article.

Conflicts of interest

The authors declare that they have no conflicts of interest.

References

[1] A.A. Adam, F. M. Mahomed, Integration of Ordinary Differential Equations via Nonlocal Symmetries, Nonlinear Dynamics, Springer, Cham 30(2002), no. 3, 267–275. Doi: 10.1023/a:1020518129295

[2] D.J. Arrigo, Symmetry Analysis of Differential Equations,Wiley (2014). Url: https://n9.cl/plyqs [3] T. Aziz, A. Fatima, C.M. Khalique, F.M. Mahomed, Prandtl’s boundary

layer equation for two-dimensional flow: Exact solutions via the simplest equation method, Mathematical Problems in Engineering, Hindawi, 1(2013), no. 724385, 1–5. Doi: 10.1155/2013/724385

[4] W.G. Bickley, LXXIII. The plane jet, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23(1937), no. 156, 727– 731. Doi: 10.1080/14786443708561847

[5] G. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, (2002). Doi: 10.1007/b97380

[6] G. Bluman, S. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York (2008). Doi: 10.1007/b97380

[7] G.W. Bluman, A.F. Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York (2010). Doi: 10.1007/978-0-387-68028-6

[8] J.P. Boyd, The Blasius function: Computations before computers, the value of tricks, undergraduate projects, and open research problems, SIAM, Society for Industrial and Applied Mathematics, Philadelphia PA, 50(2008), no. 4, 791–804. Doi: 10.1137/070681594

[9] K.P. Burr, T.R. Akylas, C.C. Mei, Chapter Two - Dimensional Laminar Boundary Layers, SIAM, Society for Industrial and Applied Mathematics, Philadelphia PA (2010). Accessed on 5-11-

2021. Url: https:// web. mit. edu/ fluids-modules/  www/

highspeed_flows/ ver2/ bl_Chap2. pdf

[10] J. Chazy, Sur les équations différentielles dont l’integrale générale est uniforme et admet des singularités essentielles mobiles, C. R. Acad. Sc. Paris 149(1909), no. 1, 563–565.

[11] J. Chazy, Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Mathematica, International Press of Boston 34(1962), no. 0, 317–385. Doi: 10.1007/bf02393131

[12] P.A. Clarkson, P.J. Olver, Symmetry and the Chazy equation, Journal of Differential Equations, 124(1996), no. 1, 225–246. Doi: 10.1006/jdeq.1996.0008 [13] M.B. Glauert, The wall jet, Journal of Fluid Mechanics, Cambridge Uni-

versity 1(1956), no. 6, 625. Doi: 10.1017/S002211205600041X

[14] P.E. Hydon, Applications of Lie Groups to Differential Equations A Beginner’s Guide, Cambridge University Press (2000). Doi: 10.1017/CBO9780511623967

[15] N.H. Ibragimov, M. Torrisi, A. Valenti, Preliminary group classification of equations v_{tt}=f (x, v_{x})v_{xx} + g(x, v_{x}), Journal of Mathematical Physics, AIP Publishing 32(1991), no. 11, 2988. Doi: 10.1063/1.529042

[16] N.H. Ibragimov, M. Torrisi, A simple method for group analysis and its application to a model of detonation, Journal of Mathematical Physics, AIP Publishing 33(1992), no. 11, 3931–3937. Doi: 10.1063/1.529841

[17] N.H. Ibragimov, M.C. Nucci, Integration of third order ordinary differential equations by Lie’s method: Equations admitting three-dimensional Lie algebras, Lie Groups and their Applications 1(1994), no. 2, 49–64.

[18] N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, CRC Press, Boca Raton FL (1995), Vol. 3. Url: https://n9.cl/jamtq

[19] N.H. Ibragimov, N. Säfström, The equivalence group and invariant solutions of a tumour growth model, Communications in Nonlinear Science and Numerical Simulation, Elsevier 9(2004), no. 1, 61–68. Doi: 10.1016/s1007-5704(03)00015-7

[20] G. Loaiza, Y. Acevedo, O.M.L. Duque, Álgebra óptima y soluciones invariantes para la ecuación de Chazy, Ingeniería y Ciencia, Eafit University, International Journal of Differential Equations, Hindawi, 17(2021), no. 33, 7–21. Doi: 10.17230/ingciencia.17.33.1

[21] G. Loaiza, Y. Acevedo, O.M.L. Duque, D.A. García Hernández, Lie algebra classification, conservation laws, and invariant solutions for a generalization of the Levinson–Smith equation, International Journal of Differential Equations, 1(2021), no. 1, 1–11. Doi: 10.1155/2021/6628243

[22] G. Loaiza, Y. Acevedo, O.M.L. Duque, D.A. García Hernández, Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie, Revista Integración, temas de matemáticas, Universidad Industrial de Santander 39(2021), no. 2, 1–11. Doi: 10.18273/revint.v39n2-2021007 [23] R. Naz, F M. Mahomed, D.P. Mason, Symmetry solutions of a third-

order ordinary differential equation which arises from Prandtl boundary layer equations, Journal of Nonlinear Mathematical Physics, Atlantis Press 15(2008), no. sup 1, 179–191. Doi: 10.2991/jnmp.2008.15.s1.16

[24] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1986). Doi: 10.1007/978-1-4684-0274-2

[25] L.V. Ovsiannikov, Group Analysis of Differential Equations, Nauka, Moscow, 1978. English transl., Academic Press, New York (1982). Url: https://www.elsevier.com/books/group-analysis-of-differential-equations/ovsiannikov/978-0-12-531680-4

[26] N. Riley, Asymptotic expansions in radial jets, Journal of Mathematics and Physics, Wiley 41(1962), no. 1-4, 132–146. Doi: 10.1002/sapm1962411132

[27] H. Schlichting, Laminare Strahlausbreitung, ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, Wiley, 13(1933), no. 2, 260–263. Doi: 10.1002/zamm.19330130403

[28] H.B. Squire, Radial Jets, Springer, Book: 50 Jahre Grenzschichtforschung (1955), 47–54. Doi: 10.1007/978-3-663-20219-6_5